We introduce the graded bialgebra deformations, which explains Andruskiewitsch-Schneider’s liftings method. We also relate this graded bialgebra deformation with the corresponding graded bialgebra cohomology groups, which is the graded version of the one due to Gerstenhaber-Schack.

We construct the graded q-differential algebra on a ZN -graded algebra by means of a graded q-commutator. We apply this construction to a reduced quantum plane and study the first order differential calculus on a reduced quantum plane induced by the N -differential of the graded q-differential algebra. 1 Graded q-differential algebra In this section given a ZN -graded algebra we construct the g...

Graded guidance labels are widely used in neural map formation, but it is not well understood which potential strategy leads to their graded expression. In midbrain tectal map development, FGFs can induce an entire midbrain, but their protein distribution is unclear, nor is it known whether they may act instructively to produce graded gene expression. Using a receptor-alkaline phosphatase fusio...

The main goal of this article is to explore the concepts graded ϕ-2-absorbing and primary submodules as a new generalization 2-absorbing submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be function, where GS(M) denotes collection R-submodules M. A proper K∈GS(M) said R-submodule M if whenever x,y are homogeneous elements R s element with xys∈K−ϕ(K), then xs∈K or ys∈K xy∈(K:RM), we call K xs ys in radical xy∈(...

If D is a tame central division algebra over a Henselian valued field F , then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF . After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map ...

We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W -polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W -polynomials of sign-graded posets has the right sign at...